Let $A$ be the $0\times 0$ matrix and let $\det(A)$ be its determinant. I am wondering if $\det(A)$ should be defined as $0$ or $1$.
- If we use the definition that determinant of an $n\times n$ matrix $(a_{ij})$ be defined as $\displaystyle\sum(-1)^{\tau(j_1\cdots j_n)}a_{1j_1}\cdots a_{n{j_n}}$, where $\tau(j_1\cdots j_n)$ is the inversion number of the permutation $j_1\cdots j_n$, then since there is no term present, $\det(A)$ should be defined as $0$.
- If we expand the $1\times 1$ matrix $(1)$ along the first row, we obtain
$$1=\det(1)=1\cdot\det(A),$$ which implies that $\det(A)$ should be defined as $1$.
Which definition of the determinant of the $0\times 0$ matrix $A$, if any, makes more sense here?